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Existence of weak solutions to stochastic differential equations in the plane with continuous coefficients


Author: J. Yeh
Journal: Trans. Amer. Math. Soc. 290 (1985), 345-361
MSC: Primary 60H10
DOI: https://doi.org/10.1090/S0002-9947-1985-0787969-7
MathSciNet review: 787969
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Abstract: Let $ B$ be a $ 2$-parameter Brownian motion on $ {\mathbf{R}}_ + ^2$. Consider the nonMarkovian stochastic differential system in $ 2$-parameter

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {dX(z) = \alpha (z,X)\;dB(z) + \... ...text{for}}\;z \in \partial {\mathbf{R}}_ + ^2,} \hfill \\ \end{array} } \right.$

i.e.,

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {X(z) = X(0) + \int_{{R_z}} {\al... ...}}_ + ^2,} \hfill \\ {x(0) = \xi ,} \hfill & {} \hfill \\ \end{array} } \right.$

where $ {R_z} = [0,s] \times [0,t]$ for $ z = (s,t) \in {\mathbf{R}}_ + ^2$. An existence theorem for weak solutions of the system is proved in this paper. Under the assumption that $ \alpha $ and $ \beta $ satisfy a continuity condition and a growth condition and $ {\mathbf{E}}[{\xi ^6}] < \infty $, it is shown that there exist a $ 2$-parameter stochastic process $ X$ and a $ 2$-parameter Brownian motion $ B$ on some probability space satisfying the stochastic integral equation above, with $ X(0)$ having the same probability distribution as $ \xi $.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0787969-7
Keywords: Brownian sheet, stochastic differential equations, tightness, Kolmogorov condition
Article copyright: © Copyright 1985 American Mathematical Society

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